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With a single explanatory variable, recover the correlation coefficient from the R2 that the linear regression gives.

FRM® Part I · Quantitative Analysis · Linear Regression · FRM1.QA.7.h

The idea

In a regression with a single explanatory variable, R² is the square of the sample correlation between X and Y. So the absolute correlation is the square root of R². The sign of the correlation is the sign of the estimated slope, because of this link: Slope = correlation × (σY ÷ σX). R² itself is the fraction of the variation in Y explained by the regression, explained sum of squares (ESS) divided by total sum of squares (TSS). Total variation splits into explained and residual (RSS) parts, so the sums of squares add up: TSS = ESS + RSS. An R² of 0.36 with a positive slope means a correlation of +0.60, not 0.36. The same R² results whichever variable is regressed on the other. The usual mistake is dropping the sign or forgetting to take the square root. This shortcut works only with one regressor. With several, R² is not the square of a single correlation.

Formulas

Slope from the correlation with one regressor

b = ρ × (σY ÷ σX)

ρ
the correlation between X and Y, ± √R²
σY
the standard deviation of Y
σX
the standard deviation of X

One explanatory variable only. The sign of ρ is the sign of the slope; the traps are dropping the sign or forgetting the square root of R².

A practice question

From the question bank, with the reason each wrong option is wrong.

In a regression of Y on a single explanatory variable X, R² is equal to:

  1. the sample correlation between X and YWhy not: R² is the square of the correlation; without squaring it would be the correlation itself.
  2. the slope coefficient divided by its standard errorWhy not: That is the t-statistic, which has no upper bound of 1.
  3. the squared sample correlation of X and Y - correct
  4. one minus the correlation between X and YWhy not: No such relation exists; R² is the squared correlation.

With one regressor, R² = (correlation)², the explained fraction of the variation in Y.

A harder one

In a regression of bond price changes on yield changes, R² is 0.30 and the estimated slope is -0.80. The correlation between the two variables is closest to:

  1. 0.55Why not: Takes the square root but drops the sign; the correlation has the same sign as the slope.
  2. -0.30Why not: Gives R² a negative sign without taking the square root.
  3. -0.80Why not: Reports the slope as the correlation; the slope is in units of Y per unit of X and depends on the standard deviations.
  4. -0.55 - correct

|correlation| = sqrt(0.30) = 0.548, and its sign is the sign of the slope (negative), so correlation = -0.55.

The hardest tier

A regression of Y on X gives R² of 0.36 and a slope of 0.90. A colleague regresses X on Y using the same data. The slope in the colleague's regression is closest to:

  1. 1.11Why not: Takes the reciprocal of 0.90; the reverse slope is not the inverse unless the fit is perfect.
  2. 0.32Why not: Multiplies R² by the slope instead of dividing.
  3. 0.67Why not: Divides the correlation 0.60 by the slope; the product of the two slopes is R², not the correlation.
  4. 0.40 - correct

The product of the two slopes equals the squared correlation, R²: b(X on Y) = 0.36 / 0.90 = 0.40. The correlation is +0.60 in both regressions.

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